The exact description of the Floer chain complex will require some additional setup, and we refer the reader to Section  for a precise statement. For now we will content ourselves with some homological results that directly follow from the description. As \(b_1(M_{\varphi })=1\) , there are interesting local coefficient systems on the configuration space. Recall for example that to each a 1-cycle \(\eta \)  in \(M_{\varphi }\) , in [1]} the authors associate a local system \(\Gamma _{\eta }\)  with fiber \(\mathbb {R}\) . The corresponding Floer groups with local coefficients \(\) HM\(\) 

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\(\widehat{\phantom{\mathit {HM}}}\) 
 HM
HM(M,sL;)\( are \) R[U]\(-modules, finite dimensional as \) R\(-vector spaces, which play a central role when discussing gluing formulas for Seiberg-Witten invariants \cite [Section 3.8]{KM}. We will begin by stating the result for such local systems \)\( as they are simpler to state.\) We will see in Section  that, because \(\Sigma /\varphi =\mathbb {P}^1\) , \(\zeta \)  and \(\bar{\zeta }\)  are never simultaneously eigenvalues of \(\tilde{\varphi }^*\) , so that we can write the list of eigenvalues (which are elements in \(S^1\) ) as
\(e^{ i\pi r_1},\dots , e^{ i\pi r_k}\quad \text{with}\quad 0<|r_1|<\dots <|r_k|<1,\) 
